Substring

This article is about the definition of a substring. For the computer function which performs this operation, see String functions (programming).

Not to be confused with subsequence, a generalization of substring.

"string" is a substring of "substring"

In formal language theory and computer science, a substring is a contiguous sequence of characters within a string.^{[citation needed]} For instance, "the best of" is a substring of "It was the best of times". In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring.

Prefixes and suffixes are special cases of substrings. A prefix of a string ${\displaystyle S}$ is a substring of ${\displaystyle S}$ that occurs at the beginning of ${\displaystyle S}$; likewise, a suffix of a string ${\displaystyle S}$ is a substring that occurs at the end of ${\displaystyle S}$.

The substrings of the string "apple" would be: "a", "ap", "app", "appl", "apple", "p", "pp", "ppl", "pple", "pl", "ple", "l", "le" "e", "" (note the empty string at the end).

Substring

A string ${\displaystyle u}$ is a substring (or factor)^[1] of a string ${\displaystyle t}$ if there exists two strings ${\displaystyle p}$ and ${\displaystyle s}$ such that ${\displaystyle t=pus}$. In particular, the empty string is a substring of every string.

Example: The string ${\displaystyle u=}$ana is equal to substrings (and subsequences) of ${\displaystyle t=}$banana at two different offsets:

banana
 |||||
 ana||
   |||
   ana

The first occurrence is obtained with ${\displaystyle p=}$b and ${\displaystyle s=}$na, while the second occurrence is obtained with ${\displaystyle p=}$ban and ${\displaystyle s}$ being the empty string.

A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix; for example, nan is a prefix of nana, which is in turn a suffix of banana. If ${\displaystyle u}$ is a substring of ${\displaystyle t}$, it is also a subsequence, which is a more general concept. The occurrences of a given pattern in a given string can be found with a string searching algorithm. Finding the longest string which is equal to a substring of two or more strings is known as the longest common substring problem. In the mathematical literature, substrings are also called subwords (in America) or factors (in Europe). ^{[citation needed]}

Prefix

See also: String operations § Prefixes

A string ${\displaystyle p}$ is a prefix^[1] of a string ${\displaystyle t}$ if there exists a string ${\displaystyle s}$ such that ${\displaystyle t=ps}$. A proper prefix of a string is not equal to the string itself;^[2] some sources^[3] in addition restrict a proper prefix to be non-empty. A prefix can be seen as a special case of a substring.

Example: The string ban is equal to a prefix (and substring and subsequence) of the string banana:

banana
|||
ban

The square subset symbol is sometimes used to indicate a prefix, so that ${\displaystyle p\sqsubseteq t}$ denotes that ${\displaystyle p}$ is a prefix of ${\displaystyle t}$. This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

Suffix

A string ${\displaystyle s}$ is a suffix^[1] of a string ${\displaystyle t}$ if there exists a string ${\displaystyle p}$ such that ${\displaystyle t=ps}$. A proper suffix of a string is not equal to the string itself. A more restricted interpretation is that it is also not empty.^[1] A suffix can be seen as a special case of a substring.

Example: The string nana is equal to a suffix (and substring and subsequence) of the string banana:

banana
  ||||
  nana

A suffix tree for a string is a trie data structure that represents all of its suffixes. Suffix trees have large numbers of applications in string algorithms. The suffix array is a simplified version of this data structure that lists the start positions of the suffixes in alphabetically sorted order; it has many of the same applications.

Border

A border is suffix and prefix of the same string, e.g. "bab" is a border of "babab" (and also of "baboon eating a kebab").^{[citation needed]}

Superstring

A superstring of a finite set ${\displaystyle P}$ of strings is a single string that contains every string in ${\displaystyle P}$ as a substring. For example, ${\displaystyle {\text{bcclabccefab}}}$ is a superstring of ${\displaystyle P=\{{\text{abcc}},{\text{efab}},{\text{bccla}}\}}$, and ${\displaystyle {\text{efabccla}}}$ is a shorter one. Concatenating all members of ${\displaystyle P}$, in arbitrary order, always obtains a trivial superstring of ${\displaystyle P}$. Finding superstrings whose length is as small as possible is a more interesting problem.

A string that contains every possible permutation of a specified character set is called a superpermutation.

See also

• Brace notation
• Substring index
• Suffix automaton

References

1. Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.
2. Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
3. Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.